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The main property of gyroscopic devices is maintaining the axis of a spinning rotor, a mathematical model formulated on the principle of the change in the angular momentum. This principle is used for mathematical modeling of the motions of a top at known publications. Nevertheless, practical tests of gyroscopic devices do not correspond to this analytical approach. Recent investigations have demonstrated that the origin of gyroscope properties is more complex than that represented in known publications. The applied torque on a gyroscope produces internal torques of the spinning rotor based on the action of the several inertial forces. These forces are the centrifugal, Coriolis, and common inertial forces as well as the change in the angular momentum generated by the mass elements and center-mass of the spinning rotor. The action of these torques manifests itself in the resistance and precession torques of the gyroscopic devices. These inertial torques act simultaneously and interdependently around two axes and represent the fundamental principles of the gyroscope theory. The new inertial torques enable deriving mathematical models for the motions of well-known top that is the simplest form of gyroscopic devices. The novelty of the work is mathematical models for the motions of the top based on action of eight inertial forces acting around its two axes. The obtained mathematical models for the top nutation and self-stabilization are represented in terms of machine dynamics and vibration analysis. The new analytical approach for motions of the well-balanced top and top with eccentricity of the center-mass definitely responds to the practical results.

L. Euler first laid out the mathematical foundations for the gyroscope theory in his work on the dynamics of rigid bodies back in 1765. Since Industrial Revolution time, I. Newton, J-L. Lagrange, L. Poinsot, J. L. R. D’Alembert, P-S. Laplace, L. Foucault, and other brilliant scientists have investigated, developed, and added new interpretations of the gyroscope effects, which are in full display in the rotor’s persistence in maintaining its plane of rotation. The applied theory of gyroscopes emerged mainly in the twentieth century in numerous publications that described the gyroscope effects [

Mechanically, a gyroscope is a spinning disc in which the axle is free to assume any orientation. The simplest gyroscope is represented by a top toy that is one of the most remarkable and widely recognized toys in the world. They still attract attention through their astonishing behavior and unusual gyroscopic properties. The motions of top toy have been described analytically in numerous publications and with complex numerical modeling [

However, all publications contain approximations, assumptions, and simplifications and explain gyroscope effects by the physical principle of the spinning rotor’s angular momentum that generates the precession torque and motions. Gyroscope effects still represent a problem that remains to be solved. The origin of gyroscope effects is more complex than those represented in the theories known to date. Recent investigations into the physical principles of gyroscope motions demonstrate four classical inertial forces acting upon a spinning rotor to generate the gyroscope effects. Research shows that centrifugal, common inertial, and Coriolis forces, in addition to change in the angular momentum of spinning rotors, are the basis for all gyroscope effects and properties [

Internal torques acting on a gyroscope.

Type of a torque generated by | Equation (N·m) |
---|---|

Centrifugal forces, | |

Inertial forces, | |

Coriolis forces, | |

Change in an angular momentum, | |

Resistance torque, | |

Precession torque, | |

In Table

When applied to a gyroscope, an external torque generates angular velocities of precessions around two axes and activates simultaneously and interdependently eight internal torques. The equations of these torques (Table

The new mathematical models for gyroscope motions will consider the simultaneous and interdependent actions of internal torques (Table

Observation of a fast-spinning top shows that it preserves the vertical position of the axis, remains steady on the supporting pivot, and avoids tilting or falling to the ground. If the axis of a spinning top toy is inclined from the vertical axis under the action of an external torque, the axis starts to describe the vertical circular cone of the precessed motion. Then the spinning top stabilizes itself and its spin axis goes to a vertical position.

However, the axis of a spinning top has free oscillation of the periodic complex motion, which is called nutation, and reveals itself as fast shivering of the precessing axis. This phenomenon of the top’s gyroscope nutation represents small but fast oscillations of the spinning rotor’s axis at about its mean position. The nature of gyroscope nutation is represented by the following sources: the center-mass is displaced from the axis of rotation, the shocked displacement of the center-mass is generated by the action of an external force and the imperfect surface geometry of the tip of the top’s leg, and so forth.

The action of the center-mass, having been displaced from the spinning top, represents the continuing existence of a disturbing force that generates a forced oscillation. Any free oscillation of the spinning top is connected with an angular acceleration that eventually ceases because of a loss of energy. The amplitudes of free oscillated motions are very small in the case of a rapidly rotating top. Because of the inevitable presence of resistances, these oscillations for a well-balanced top toy are subject to asymptotically rapid decay. In most cases, the nutation of a well-balanced top toy is quickly damped by the action of inertial resistance forces and friction forces in the bearing, leaving uniform precession.

Gyroscopes with a high angular velocity represent overdamped or critically damped systems according to the theory of vibration [

The nutation process of the top can be considered in terms of machine dynamics, particularly in terms of vibration analysis [

Vibration analysis often takes into account the energy losses by means of a single factor called the damping factor. The nutation model of the spinning top consists of a not so well-balanced mass and a damper that represents the action of the top’s weight and several internal torques. In the case of a gyroscope’s free nutation process, the action of the single shock type torque on a spinning top toy represents one period of an oscillation with a short time for a single cycle. The damping resistance torques decay the action of one shock torque, resulting in the inclination of only the spinning top’s axis. For well-balanced gyroscopic devices and a top with high angular velocities, the process of the damped oscillation is not displayed.

For a top that possesses an eccentric rotating mass, that is, the center-mass being displaced on some small distance from the top’s axis, nutation is a regular process. The centrifugal force of the offset mass is asymmetric, which causes an angular displacement of the top’s shaft. The spinning top toy is constantly being displaced and moved by these asymmetric forces. The eccentric rotating mass of the top represents harmonic nutation, which means that the top is forced to nutate at the frequency of excitation [

The position of a top’s eccentric motion according to sinusoidal law.

The analysis of a gyroscope motions and nutation is conducted using the example of a top that is not well balanced and is thus tilted (Figure

Forces acting on a spinning top.

As a result, the top’s axis will begin to precess at about the vertical and horizontal positions. The precessions of the top are also accompanied by visually undetectable nutational oscillations, which decay rapidly according to the action of the internal torques. Because of the internal torques’ resistance forces, the proper rotation of the top gradually slows down, while the precession velocities correspondingly increase. When the angular velocity of the top becomes smaller, the resistance forces of the internal torques become weaker, and the top becomes unstable and falls. In the case of a slowly rotating top, the nutational oscillations are noticeable and manifest a substantial change in the pattern of the top toy’s axial movement. In this case, the end of the top’s axis can describe a clearly visible wavy or looped curve alternately departing from the vertical position. In the case of a highly rotating top, the nutational oscillations can decrease and manifest the stabilization of the top’s axial movement. The top’s axis can approach the vertical position.

The motions of a not so well-balanced spinning top are complex. The equation of nutation is formulated by the action of several forces on the top. These forces represent the top’s weight, the centrifugal forces of eccentric mass and center-mass, the centrifugal and Coriolis forces generated by the rotating mass elements, the common inertial forces, and the rate of change in the angular momentum of the spinning toy. These forces produce the torques acting on the top. These are demonstrated in Figure

The centrifugal force of the eccentric mass generates the variable torque represented by the following equation [

The weight of a top generates torque, for which the equation is as follows:

The parameters defined above allow for the formulation of a mathematical model for a top’s motion around axes

The torques generated by the centrifugal forces of the rotating top’s center-mass around axis

The methodology towards a solution for (

Solving (

The example considers the nutation process of a disc-type top whose data is presented in Table

Technical data of a top.

Parameter | Data |
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Angular velocity, | 2000 rpm |

Radius of the disc, | 0.025 m |

Length of the leg, | 0.02 m |

Eccentricity of the centre mass, | 0.0001 m |

Angle of tilt, | 75.0° |

Weight, | 0.02 kg |

Mass moment of inertia, kgm^{2} | |

Around axis | 0.625 × 10^{−5} |

Around axes | 0.3125 × 10^{−5} |

Around axes | 1.1125 × 10^{−5} |

The torque generated by the top’s weight is as follows:

Substituting the defined parameters (Table

Substituting the new expressions into (

Solving (

Equation (

The time spent on a half oscillation is as follows:

The maximal and minimal value of the linear velocities of the top center-mass around axis

The maximal and minimal value of the amplitudes of the center-mass nutation around

This diagram of the top’s nutation loop is presented in Figure

Nutation’s amplitudes of top’s center-mass.

The magnitudes of precessed motion’s distance (

Deployed diagram of top’s nutation.

The deployed diagram of top’s nutation can have loops if the negative amplitude of oscillation along axis

The right component is bigger than the left one, that is, the top stabilizes itself.

The simplest form of a gyroscope is a top, whose motions are described by known publications in terms of mathematical models based on the change in the angular momentum. The analytical study of forces acting on a well-balanced top and on a top toy with the eccentricity of center-mass has formulated new mathematical models for its motions. These models are based on action of several internal forces generated by the mass elements and center-mass of a top. The action of a top’s weight produces its own internal torques that interrelate and act at one time and expresses precession motions. A top rotating eccentric center-mass manifests a nutation process. The obtained mathematical models for the top’s motions enable describing physical principles of acting forces. These models of motions for a well-balanced top allowed its minimal angular velocity for self-stabilization to be defined. The diagram of a top’s nutation represents the motions of the top with eccentricity of its center-mass. The new mathematical models for the motions of the well-balanced top and with eccentricity of its center-mass definitely respond to the practical results.

New studies of the gyroscope effects have shown that the origin of the acting forces and motions in a gyroscope is more complex. The gyroscope effects result from action of the several inertial torques generated by the centrifugal, Coriolis, and common inertial forces as well as the change in the angular momentum of the spinning rotor. The action of the new internal torques clearly describes the physics of gyroscope’s motions and changes the traditional presentation of gyroscope effects. This new study enables describing gyroscope properties that were unexplainable at former time. Today new mathematical models for acting internal torques can be applied to any gyroscopic devices and manually solve all technical problems. These new analytical solutions were applied for describing the forces acting on a well-balanced top and on a top with the eccentricity of center-mass. The new physical principles enabled formulating mathematical models of top processed motions, nutation, and its ability to rotate vertically.

In gyroscope theory, the top’s motions are one of the most complex and intricate in terms of analytical solutions. Known mathematical models for the top motions are accepted with simplifications and do not adequately express a real picture of its motions. The new mathematical models for gyroscope torques consider the simultaneous and interdependent action of several inertial forces generated by the rotating mass of the spinning rotor. As a practical application, these new physical principles for gyroscope motions were used for modeling a top’s motions that include micro oscillation and self-stabilization. These mathematical models are thus distinguishable from those in well-known publications, which tend to have complex numerical modeling that does not interpret the origin of gyroscope effects. The application of new mathematical models for the top’s motions effectively and clearly demonstrates physical principles of acting forces and motions. In that regard, this is also a good example of educational processes.

Linear motion along axis

Maximal and minimal linear motion along axis

Gravity acceleration

Centrifugal force

Centrifugal force of a top toy’s center-mass rotating around axis

Index for axis

Mass moment of inertia of a top

Mass moment of inertia of a top around axis

Length of a top’s leg

Mass of a top

Load torque

Torque generated by the change in the angular momentum, centrifugal, Coriolis, and common inertial forces acting around axis

Resistance and precession torque acting around axis

Torques generated by eccentricity of the center-mass and action around axes

Linear velocity along axis

Maximal and minimal value of the linear velocity along axis

Time

Angle for calculating the maximal value of the torques’ magnitude

Tilt angle of a top’s axis

Angular velocity of a top

Angular velocity of precession around axis

Maximal and minimal value of the angular velocity of precession around axis

The authors declare that they have no conflicts of interest.